\section{Insurer Loss Distributions Vary With Portfolio Size} 
\label{sec:InsurerLossDistributionsVaryWithPortfolioSize} 

Two insurers, $\textbf{\textit{M}}$ and $\textbf{\textit{N}}$, randomly selecting $M$ and $N$ ($M >> N$) policyholders from the same population, with individual policyholder PLRE standard deviation, $\sigma$, and Population Loss Ratio, $\mu$, draw PLREs from very different, normally distributed Cumulative PLRE Distribution Functions: $\Phi_{M}$($\mu$,$\frac{\sigma}{\sqrt{M}}$) and $\Phi_{M}$($\mu$,$\frac{\sigma}{\sqrt{N}}$), where $\frac{\sigma}{\sqrt{M}}$ $<<$ $\frac{\sigma}{\sqrt{N}}$ and each insurer's standard error determines all other insurers' standard errors since $\sigma_{e_{M}}$ = $\sigma$ * $\frac{\sqrt{N}}{\sqrt{M}}$. 

I will specify the Population Loss Ratio (PLR) and a reasonable, market appropriate, assumption about Population Loss Ratio Estimate (PLRE) variation for a single, ``reasonably efficient'' Paradigm Insurer ($PI$), then analyze, and compare, the impact of portfolio size on operating results for four other insurers. The ``Market Premium'' for this reasonably efficient insurer is an adequate, but not excessive, expense, risk, and profit loaded premium such that if all policyholders pay the ``Market Premium,'' the expected value of the total industry PLRE equals the PLR, and the market will continue to operate. Reasonable efficiency means that when the Paradigm Insurer receives the market premium, it can continue to operate with minimal risk of insolvency and acceptable levels of profits.
